Evaluate the following limits :
`Lim_(xto0) (sqrt((1+x+x^(2)))-1)/x`

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{\sqrt{1 + x + x^2} - 1}{x}, \]

we can follow these steps:

### Step 1: Substitute \(x = 0\)

First, we substitute \(x = 0\) into the expression to check if we get an indeterminate form:

\[ \frac{\sqrt{1 + 0 + 0^2} - 1}{0} = \frac{\sqrt{1} - 1}{0} = \frac{1 - 1}{0} = \frac{0}{0}. \]

Since we have the indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule or simplify the expression.

### Step 2: Multiply by the Conjugate

To simplify the expression, we multiply the numerator and the denominator by the conjugate of the numerator:

\[ \frac{\sqrt{1 + x + x^2} - 1}{x} \cdot \frac{\sqrt{1 + x + x^2} + 1}{\sqrt{1 + x + x^2} + 1}. \]

This gives us:

\[ \frac{(\sqrt{1 + x + x^2} - 1)(\sqrt{1 + x + x^2} + 1)}{x(\sqrt{1 + x + x^2} + 1)} = \frac{(1 + x + x^2) - 1}{x(\sqrt{1 + x + x^2} + 1)}. \]

### Step 3: Simplify the Numerator

Now, simplify the numerator:

\[ 1 + x + x^2 - 1 = x + x^2. \]

So, we can rewrite our limit as:

\[ \lim_{x \to 0} \frac{x + x^2}{x(\sqrt{1 + x + x^2} + 1)}. \]

### Step 4: Factor out \(x\)

We can factor \(x\) out of the numerator:

\[ \lim_{x \to 0} \frac{x(1 + x)}{x(\sqrt{1 + x + x^2} + 1)}. \]

Now, we can cancel \(x\) from the numerator and denominator (provided \(x \neq 0\)):

\[ \lim_{x \to 0} \frac{1 + x}{\sqrt{1 + x + x^2} + 1}. \]

### Step 5: Substitute \(x = 0\) Again

Now we substitute \(x = 0\):

\[ \frac{1 + 0}{\sqrt{1 + 0 + 0^2} + 1} = \frac{1}{\sqrt{1} + 1} = \frac{1}{1 + 1} = \frac{1}{2}. \]

### Final Answer

Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{\sqrt{1 + x + x^2} - 1}{x} = \frac{1}{2}. \]

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